Complex Numbers to Complex Powers
Date: 10/19/2000 at 20:24:08 From: Michael Lee Subject: Non-e base with imaginary exponent I understand Euler's discovery of e^ix and its relation to the sine and cosine functions. But since e is just a number (although a special one), what is the general value of n^i? Specifically, since a real number to a real power equals a real, does a real number to a complex power equal some complex number? If so, what is it? Thank you. Michael Lee
Date: 10/19/2000 at 22:47:37 From: Doctor Peterson Subject: Re: Non-e base with imaginary exponent Hi, Michael. You're right; in fact, we can use the formula to raise any complex number to any complex power. Any real number a can be written as e^ln(a); so a^(ix) = (e^ln(a))^(ix) = e^(ix*ln(a)) = cos(x*ln(a)) + i*sin(x*ln(a)) We can extend this to complex exponents this way: a^(x+iy) = a^x * a^(iy) To allow for complex bases, write the base in the form a*e^(ib), and you find [a*e^(ib)]^z = a^z * e^(ib*z) These ideas will allow you to raise any real or complex base to any real or complex exponent. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/20/2000 at 15:42:38 From: Michael Lee Subject: Real raised by imaginary number Doctor Peterson, I asked what the result was of raising a real by a complex number. You sent me a great reply, which is understandable on an analytical level. However, when viewed geometrically, it seems that result of this operation maps onto a unit circle on the complex plane. This seems too fantastic. I've relied on intuition for my sense of mathematics, but this is beyond my senses. How is it possible that all combinations of real numbers raised to powers with only an imaginary component degenerate to the unit circle? If I had to make up an explanation I might say that in the world of numbers, reals cause translation and imaginaries cause rotation; also, since multiplying by i is often seen as rotating by 90 degrees in the complex plane, then by extension, raising by an imaginary is just a more complicated version of this rotation characteristic of i. Is this possible? Thank you.
Date: 10/20/2000 at 16:44:36 From: Doctor Peterson Subject: Re: Real raised by imaginary number Hi, Michael. Yes, you're right: any real raised to any pure imaginary power gives a complex number with absolute value 1, so the mapping y -> e^(iy) takes the real line onto the unit circle, wrapping it around with a period of 2 pi. That's what Euler's formula means: raising e to an imaginary power produces the complex number with that angle. You may want to look at our FAQ on this, Imaginary Exponents and Euler's Equation http://mathforum.org/dr.math/faq/faq.euler.equation.html When we raise a real to a complex power, the imaginary component of the exponent rotates it while the real component dilates (not translates) it. That is, the angle of the resulting number is determined by the imaginary part of the exponent, while the absolute value is determined by the real part. This is what gives Euler's formula tremendous power, and in fact this is the answer to the often- asked question, what good are complex numbers? They let us combine dilation and rotation, or exponential and sinusoidal functions, into a single operation. Multiplication by a complex number, similarly, rotates by the angle of the complex number (adding the two angles), and dilates by its absolute value (multiplying the absolute values.) - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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